The present invention relates to an electromagnetic flowmeter and, more particularly, to an electromagnetic flowmeter which detects a characteristic or state of a fluid or a state in a measuring tube and corrects the flow rate of the fluid.
An electromagnetic flowmeter is a flowmeter which can measure a flow rate with stability owing to its characteristics, and has established a position as a high-accuracy flowmeter. However, a conventional electromagnetic flowmeter is assumed to operate under a condition where the tube is filled with a fluid to be measured, and is generally designed to obtain the flow rate of the fluid by detecting a signal proportional to the flow velocity of the fluid and multiplying the flow velocity by the sectional area of the tube. If, therefore, the tube is not filled with a fluid or air bubbles are mixed in the fluid, an error occurs in flow rate measurement.
For example, as the level of a fluid in the tube varies, the sectional area varies. As a consequence, a span variation occurs in an output from the electromagnetic flowmeter, and a flow rate error occurs. In addition, when air bubbles are mixed in the tube, the volume of the fluid varies. This causes a span variation in an output, resulting in an error in the flow rate to be obtained. For this reason, in order to measure the flow rate of the fluid with high accuracy, it is necessary to perform flow rate correction by measuring a state of the fluid, e.g., the fluid level or the amount of air bubbles mixed by using another sensor. As described above, when a substance other than a fluid to be measured is mixed in the tube, for example, a gas and a liquid or the like constitute a multiphase flow (a fluid level variation can also be regarded as a case wherein air is mixed in the fluid), it is difficult to accurately measure a flow rate by using a general electromagnetic flowmeter. Under the circumstances, demands have arisen for an electromagnetic flowmeter which can accurately measure a flow rate even when a state of a fluid varies.
Against the background described above, an electromagnetic flowmeter for a partially filled tube has been proposed in, for example, reference 1 (Japanese Patent Laid-Open No. 6-241855) and reference 2 (JNMIHF edition, “Flow Rate Measurement from A to Z for Instrumentation Engineers”, Kogyo Gijutusha, 1995, pp. 147-148) for the correction of fluid level variations. These references propose a technique of correcting a flow rate by measuring a fluid level as an application of an electromagnetic flowmeter for a partially filled tube. The electromagnetic flowmeter disclosed in references 1 and 2 obtains first a fluid level from the ratio between the signal electromotive force obtained by a pair of electrodes provided on the left and right sides of a channel when exciting coils provided on the upper and lower sides of the channel are simultaneously driven and the signal electromotive force obtained when the exciting coil on the upper side is singly driven, corrects the sensitivity which has been obtained in advance on the basis of the fluid level, and outputs a flow rate.
On the other hand, the present inventor has proposed an electromagnetic flowmeter which solves the problem of a span shift in reference 3 (WO 03/027614). A physical phenomenon necessary for the explanation of reference 3 and the present invention will be described below. When an object moves in a changing magnetic field, electromagnetic induction generates two types of electric fields, namely (a) electric field E(i)=∂A/∂t which is generated by a temporal change in magnetic field, and (b) electric field E(v)=v×B which is generated as the object moves in the magnetic field. In this case, v×B represents the outer product of v and B, ∂A/∂t represents the partial differential of A with respect to time. In this case, v, B, and A respectively correspond to the following and are vectors having directions in three dimensions (x, y, and z) (v: flow velocity, B: magnetic flow density, and A: vector potential (whose relationship with the magnetic flux density is represented by B=rotA)). Note, however, that the three-dimensional vectors in this case differ in meaning from vectors on a complex plane. These two types of electric fields generate a potential distribution in the fluid, and electrodes can detect this potential.
Generally known mathematical basic knowledge will be described next. A cosine wave P·cos(ω·t) and a sine wave Q·sin(ω·t) which have the same frequency but different amplitudes are combined into the following cosine wave. Let P and Q be amplitudes, and ω be an angular frequency.P·cos(ω·t)+Q·sin(ω·t)=(P2+Q2)1/2·cos(ω·t−ε)for ε=tan−1(Q/P)  (1)
In order to analyze the combining operation in equation (1), it is convenient to perform mapping on a complex coordinate plane so as to plot an amplitude P of cosine wave P·cos(ω·t) along a real axis and an amplitude Q of the sine wave Q·sin(ω·t) along an imaginary axis. That is, on the complex coordinate plane, a distance (P2+Q2)1/2 from the origin gives the amplitude of the combined wave, and an angle e=tan−1(Q/P) gives the phase difference between the combined wave and ω·t.
In addition, on the complex coordinate plane, the following relational expression holds.L·exp(j·ε)=L·cos(ε)+j·L·sin(ε)  (2)
Equation (2) is an expression associated with a complex vector, in which j is an imaginary unit, L gives the length of the complex vector, and e gives the direction of the complex vector. In order to analyze the geometrical relationship on the complex coordinate plane, it is convenient to use conversion to a complex vector.
The following description uses mapping onto a complex coordinate plane like that described above and geometrical analysis using complex vectors to show how an inter-electrode electromotive force behaves and explain how the electromagnetic flowmeter of reference 3 and the present invention use this behavior.
A complex vector arrangement with one coil set and an electrode pair in the electromagnetic flowmeter described in reference 3 will be described next.
FIG. 36 is a block diagram for explaining the principle of the electromagnetic flowmeter in reference 3. This electromagnetic flowmeter includes a measuring tube 101 through which a fluid to be measured flows, a pair of electrodes 102a and 102b which are placed to face each other in the measuring tube 101 so as to be perpendicular to both a magnetic field to be applied to the fluid and an axis PAX of the measuring tube 101 and come into contact with the fluid, and detect the electromotive force generated by the magnetic flow and the flow of the fluid, and an exciting coil 103 which applies, to the fluid, a time-changing magnetic field asymmetric on the front and rear sides of the measuring tube 101 which are bordered on a plane PLN which includes the electrodes 102a and 102b, with the plane PLN serving as a boundary of the measuring tube 101.
Of a magnetic field Ba generated by the exciting coil 103, a magnetic field component (magnetic flux density) B1 orthogonal to both an electrode axis EAX connecting the electrodes 102a and 102b and the measuring tube axis PAX on the electrode axis EAX is given byB1=b1·cos(ω0·t−θ1)  (3)
In equation (3), b1 is the amplitude of the magnetic flux density B1, ω0 is an angular frequency, and θ1 is a phase difference (phase lag) from ω0·t. The magnetic flux density B1 will be referred to as the magnetic field B1 hereinafter.
An inter-electrode electromotive force which originates from a change in magnetic field and is irrelevant to the flow velocity of a fluid to be measured will be described first. Since the electromotive force originating from the change in magnetic field depends on a time derivative dB/dt of the magnetic field, and hence the magnetic field B1 generated by the exciting coil 103 is differentiated according todB1/dt=−ω0·b1·sin(ω0·t−θ1)  (4)
If the flow velocity of the fluid to be measured is 0, a generated eddy current is only a component originating from a change in magnetic field. An eddy current I due to a change in the magnetic field Ba is directed as shown in FIG. 37. Therefore, an inter-electrode electromotive force E which is generated by a change in the magnetic field Ba and is irrelevant to the flow velocity is directed as shown in FIG. 37 within a plane including the electrode axis EAX and the measuring tube axis PAX. This direction is defined as the negative direction.
At this time, the inter-electrode electromotive force E is the value obtained by multiplying a time derivative −dB1/dt of a magnetic field whose direction is taken into consideration by a coefficient k (a complex number associated with the conductivity and permittivity of the fluidity to be measured and the structure of the measuring tube 101 including the layout of the electrodes 102a and 102b), as indicated by the following equation:E=k·ω0·b1·sin(ω0·t−θ1)  (5)
Equation (5) is rewritten to the following equation:
                                                        E              =                            ⁢                                                                    k                    ·                    ω0                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              sin                        ⁡                                                  (                                                      -                            θ1                                                    )                                                                    }                                        ·                                          cos                      ⁡                                              (                                                  ω0                          ·                          t                                                )                                                                                            +                                                                                                      ⁢                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          cos                      ⁡                                              (                                                  -                          θ1                                                )                                                              }                                    ·                                      sin                    ⁡                                          (                                              ω0                        ·                        t                                            )                                                                                                                                              =                            ⁢                                                                    k                    ·                    ω0                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              -                                                  sin                          ⁡                                                      (                            θ1                            )                                                                                              }                                        ·                                          cos                      ⁡                                              (                                                  ω0                          ·                          t                                                )                                                                                            +                                                                                                      ⁢                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          cos                      ⁡                                              (                        θ1                        )                                                              }                                    ·                                      sin                    ⁡                                          (                                              ω0                        ·                        t                                            )                                                                                                                              (        6        )            
In this case, if equation (6) is mapped on the complex coordinate plane with reference to ω0·t, a real axis component Ex and an imaginary axis component Ey are given by
                                                        Ex              =                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          -                                              sin                        ⁡                                                  (                          θ1                          )                                                                                      }                                                                                                                          =                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          cos                      ⁡                                              (                                                                              π                            /                            2                                                    +                          θ1                                                )                                                              }                                                                                                          (        7        )                                                                    Ey              =                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          cos                      ⁡                                              (                        θ1                        )                                                              }                                                                                                                          =                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          sin                      ⁡                                              (                                                                              π                            /                            2                                                    +                          θ1                                                )                                                              }                                                                                                          (        8        )            
In addition, Ex and Ey represented by equations (7) and (8) are rewritten to a complex vector Ec represented by
                                                        Ec              =                            ⁢                              Ex                +                                  j                  ·                  Ey                                                                                                        =                            ⁢                                                                    k                    ·                    ω0                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              cos                        ⁡                                                  (                                                                                    π                              /                              2                                                        +                            θ1                                                    )                                                                    }                                                                      +                                                                                                      ⁢                                                j                  ·                  k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          sin                      ⁡                                              (                                                                              π                            /                            2                                                    +                          θ1                                                )                                                              }                                                                                                                                          =                                ⁢                                                                            k                      ·                      ω0                      ·                      b                                        ⁢                                                                                  ⁢                                          1                      ·                                              {                                                  cos                          ⁡                                                      (                                                                                          π                                /                                2                                                            +                              θ1                                                        )                                                                          }                                                                              +                                      j                    ·                                          sin                      ⁡                                              (                                                                              π                            /                            2                                                    +                          θ1                                                )                                                                                                        }                                                                          =                            ⁢                                                k                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                  exp                                ⁢                                  {                                      j                    ·                                          (                                                                        π                          /                          2                                                +                        θ1                                            )                                                        }                                                                                        (        9        )            
In addition, the coefficient k described above is rewritten to a complex vector to obtain the following equation:
                                                        k              =                            ⁢                                                rk                  ·                                      cos                    ⁡                                          (                      θ00                      )                                                                      +                                  j                  ·                  rk                  ·                                      sin                    ⁡                                          (                      θ00                      )                                                                                                                                              =                            ⁢                              rk                ·                                  exp                  ⁡                                      (                                          j                      ·                      θ00                                        )                                                                                                          (        10        )            
In equation (10), rk is a proportional coefficient, and θ00 is the angle of the vector k with respect to the real axis.
Substituting equation (10) into equation (9) yields an inter-electrode electromotive force Ec (an inter-electrode electromotive force which originates from only a temporal change in magnetic field and is irrelevant to the flow velocity) rewritten to complex coordinates as follows:
                                                        Ec              =                                                rk                  ·                                      exp                    ⁡                                          (                                              j                        ·                        θ00                                            )                                                        ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                  exp                                ⁢                                  {                                      j                    ·                                          (                                                                        π                          /                          2                                                +                        θ1                                            )                                                        }                                                                                                        =                                                rk                  ·                  ω0                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                  exp                                ⁢                                  {                                      j                    ·                                          (                                                                        π                          /                          2                                                +                        θ1                        +                        θ00                                            )                                                        }                                                                                        (        11        )            
In equation (11), rk·ω0·b1·exp{j·(p/2+θ1+θ00)} is a complex vector having a length rk·ω0·b1 and an angle p/2+θ1+θ00 with respect to the real axis.
An inter-electrode electromotive force originating from the flow velocity of a fluid to be measured will be described next. Letting V (V≠0) be the magnitude of the flow velocity of the fluid, since a component v×Ba originating from a flow velocity vector v of the fluid is generated in a generated eddy current in addition to the eddy current I when the flow velocity is 0, an eddy current Iv generated by the flow velocity vector v and the magnetic field Ba is directed as shown in FIG. 38. Therefore, the direction of an inter-electrode electromotive force Ev generated by the flow velocity vector v and the magnetic field Ba becomes opposite to the direction of the inter-electrode electromotive force E generated by the temporal change, and the direction of Ev is defined as the positive direction.
In this case, as indicated by the following equation, the inter-electrode electromotive force Ev originating from the flow velocity is the value obtained by multiplying the magnetic field B1 by a coefficient kv (a complex number associated with a magnitude V of the flow velocity, the conductivity and permittivity of the fluidity to be measured, and the structure of the measuring tube 101 including the arrangement of the electrodes 102a and 102b):
                              Ev          =                    ⁢                      kv            ·                          {                              b                ⁢                                                                  ⁢                                  1                  ·                                      cos                    ⁡                                          (                                                                        ω0                          ·                          t                                                -                        θ1                                            )                                                                                  }                                      ⁢                                  ⁢                              Equation            ⁢                                                                      ⁢                                                                    (            12            )                    ⁢                                          ⁢          is          ⁢                                          ⁢          rewritten          ⁢                                          ⁢          to                                    (        12        )                                                                    Ev              =                            ⁢                                                                    kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          cos                      ⁡                                              (                                                  ω0                          ·                          t                                                )                                                              ·                                          cos                      ⁡                                              (                                                  -                          θ1                                                )                                                                                            -                                                                                                      ⁢                                                kv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      sin                    ⁡                                          (                                              ω0                        ·                        t                                            )                                                        ·                                      sin                    ⁡                                          (                                              -                        θ1                                            )                                                                                                                                              =                            ⁢                                                                    kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              cos                        ⁡                                                  (                          θ1                          )                                                                    }                                        ·                                          cos                      ⁡                                              (                                                  ω0                          ·                          t                                                )                                                                                            +                                                                                                      ⁢                                                kv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                          sin                      ⁡                                              (                        θ1                        )                                                              }                                    ·                                      sin                    ⁡                                          (                                              ω0                        ·                        t                                            )                                                                                                                              (        13        )            
In this case, when mapping equation (13) on the complex coordinate plane with reference to ω0·t, a real axis component Evx and an imaginary axis component Evy are given byEvx=kv·b1·{cos(θ1)}  (14)Evy=kv·b1·{sin(θ1)}  (15)
In addition, Evx and Evy represented by equations (14) and (15) are rewritten to a complex vector Evc represented by
                                                        Evc              =                              Evx                +                                  j                  ·                  Evy                                                                                                        =                                                                    kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              cos                        ⁡                                                  (                          θ1                          )                                                                    }                                                                      +                                                      j                    ·                    kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              sin                        ⁡                                                  (                          θ1                          )                                                                    }                                                                                                                                              =                                                kv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      {                                                                  cos                        ⁡                                                  (                          θ1                          )                                                                    +                                              j                        ·                                                  sin                          ⁡                                                      (                            θ1                            )                                                                                                                }                                                                                                                          =                                                kv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      exp                    ⁡                                          (                                              j                        ·                        θ1                                            )                                                                                                                              (        16        )            
In addition, the coefficient kv described above is rewritten to a complex vector to obtain the following equation:
                                                        kv              =                                                rkv                  ·                                      cos                    ⁡                                          (                      θ01                      )                                                                      +                                  j                  ·                  rkv                  ·                                      sin                    ⁡                                          (                      θ01                      )                                                                                                                                              =                              rkv                ·                                  exp                  ⁡                                      (                                          j                      ·                      θ001                                        )                                                                                                          (        17        )            
In equation (17), rkv is a proportional coefficient, and θ01 is the angle of the vector kv with respect to the real axis. In this case, rkv is equivalent to the value obtained by multiplying the proportional coefficient rk (see equation (10)) described above by the magnitude V of the flow velocity and a proportion coefficient γ. That is, the following equation holds:rkv=γ·rk·V  (18)
Substituting equation (17) into equation (16) yields an inter-electrode electromotive force Evc rewritten to complex coordinates as follows:
                                                        Evc              =                                                kv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                                      exp                    ⁡                                          (                                              j                        ·                        θ1                                            )                                                                                                                                              =                                                rkv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                  exp                                ⁢                                  {                                      j                    ·                                          (                                              θ1                        +                        θ01                                            )                                                        }                                                                                        (        19        )            
In equation (19), rkv·b1·exp{j·(θ1+θ01)} is a complex vector having a length rkv-b1 and an angle θ1+θ01 with respect to the real axis.
An inter-electrode electromotive force Eac as a combination of inter-electrode electromotive force Ec originating from a temporal change in magnetic field and an inter-electrode electromotive force Evc originating from the flow velocity of the fluid is expressed by the following equation according to equations (11) and (19).
                                                        Eac              =                            ⁢                              Ec                +                Evc                                                                                        =                            ⁢                                                                    rk                    ·                    ω0                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                                              π                            /                            2                                                    +                          θ1                          +                          θ00                                                )                                                              }                                                  +                                                                                                      ⁢                                                rkv                  ·                  b                                ⁢                                                                  ⁢                                  1                  ·                  exp                                ⁢                                  {                                      j                    ·                                          (                                              θ1                        +                        θ01                                            )                                                        }                                                                                        (        20        )            
As is obvious from equation (20), an inter-electrode electromotive force Eac is written by two complex vectors rk·ω0·b1·exp{j·(p/2+θ1+θ00)} and rkv·b1·exp{j·(θ1+θ01)}. The length of the resultant vector obtained by combining the two complex vectors represents the amplitude of the output (the inter-electrode electromotive force Eac), and an angle φ of the resultant vector represents the phase difference (phase delay) of the inter-electrode electromotive force Eac with respect to the phase ω0·t of the input (exciting current).
Under the above principle, the electromagnetic flowmeter described in reference 3 extracts a parameter (asymmetric excitation parameter) free from the influence of a span shift, and outputs a flow rate on the basis of the extracted parameter, thereby solving the problem of the span shift.
A span shift will be described with reference to FIG. 39. Assume that the magnitude V of the flow velocity measured by the electromagnetic flowmeter has changed in spite of the fact that the flow velocity of a fluid to be measured has not changed. In such a case, a span shift can be thought as a cause of this output variation.
Assume that calibration is performed such that when the flow velocity of a fluid to be measured is 0 in an initial state (period T1), the output from the electromagnetic flowmeter becomes 0 (v), and when the flow velocity is 1 (m/sec) (period T2), the output becomes 1 (v). In this case, an output from the electromagnetic flowmeter is a voltage representing the magnitude V of a flow velocity. According to this calibration, if the flow velocity of a fluid to be measured is 1 (m/sec), the output from the electromagnetic flowmeter should be 1 (v). When a given time t1 has elapsed, however, the output from the electromagnetic flowmeter may become 1.2 (v) in spite of the fact that the flow velocity of the fluid to be measured remains 1 (m/sec). A span shift can be thought as a cause of this output variation. A phenomenon called a span shift occurs when, for example, the value of an exciting current flowing in the exciting coil cannot be maintained constant.
According to references 1 and 2, the flow rate of a fluid which flows in the tube in a partially filled state can be measured. The electromagnetic flowmeter disclosed in references 1 and 2 detects a fluid level on the basis of the ratio between the signal electromotive force obtained when the exciting coils on the upper and lower sides are simultaneously driven and the signal electromotive force obtained when the exciting coil on the upper side is separately driven. For this reason, when a signal electromotive force decreases as the flow rate approaches 0, the detected fluid level contains a large error, and the accuracy of sensitivity correction deteriorates, resulting in an flow rate measurement error.
In addition, the electromagnetic flowmeter disclosed in reference 3 can automatically perform span correction. The electromagnetic flowmeter in reference 3 can, however, perform span correction only when a parameter free from the influence of a span shift changes in the same manner as variation in span as a coefficient applied to the flow velocity.
Of the inter-electrode electromotive force detected by electrodes, a v×B component associated with the flow velocity of the fluid is represented by, for example, Ka·B·Cf·V (where Ka is a constant term, B is a term associated with a magnetic field, Cf is a term associated with a characteristic or state of the fluid, and V is the magnitude of the flow velocity), and a ∂A/∂t component associated with the characteristic or state of the fluid can be represented by, for example, Ka·B·Cg·ω (where Ka is the constant term, B is the term associated with the magnetic field, Cg is a term associated with a characteristic or state of the fluid, and X is an exciting angular frequency).
The electromagnetic flowmeter in reference 3 is assumed to keep the ratio of Cf/Cg constant. In a strict sense, a ∂A/∂t component and a v×B component change differently with a change in a characteristic or state of a fluid or state in the measuring tube, and hence the ratio of Cf/Cg is not constant. Consequently, as the required flow rate measurement accuracy increases, a flow rate measurement error occurs in the electromagnetic flowmeter in reference 3. This flow rate measurement error becomes conspicuous in particular when measurement is performed by using high-frequency excitation with an exciting current having a high frequency or when a fluid with a low conductivity is measured. Assume that the flow velocity of a fluid is accurately obtained. Even in this case, if the volume (sectional area) of the fluid varies, an error occurs in the flow rate. This may make impossible to accurately measure a flow rate even by applying the principle of reference 3.